Learning topic
Contact Strength of Machine Elements
Contact-strength assessment for bearings, rollers, gears, and wheels: Hertz pressure, pitting, fatigue life, lubrication, roughness, and edge effects.
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Contact strength is the ability of working surfaces to transmit localized contact loads without unacceptable plastic deformation, fatigue pitting, cracking, or another specified damage mode.
Contact-stress check
In a simple verification model, the calculated maximum contact pressure $p_0$ is compared with an allowable value:
$$p_0\le[p_H],$$
or an equivalent standard-specific form such as:
$$\sigma_H\le[\sigma_H].$$
- $p_0$ — calculated maximum contact pressure;
- $[p_H]$ — allowable contact pressure according to the adopted method;
- $\sigma_H$ — calculated contact stress in a standard-specific formulation;
- $[\sigma_H]$ — allowable contact stress in the same formulation.
The definitions of calculated and allowable contact stress depend on the component type and design standard; there is no single universal allowable value for all contacts.
Pitting and contact fatigue
In rolling bearings, gears, and roller pairs, contact loading repeats many times. The cyclic subsurface stress state can initiate cracks and cause local surface material removal known as pitting or spalling. Therefore, a static check based only on $p_0$ does not replace a life calculation.
Factors affecting contact strength
- Material and hardness. Heat treatment and surface hardening can strongly change resistance to contact damage.
- Geometry. Smaller radii of curvature generally increase local pressure under otherwise comparable conditions.
- Roughness. Real contact occurs through asperities that create local pressure peaks.
- Lubrication. A lubricant film can separate surfaces and alter friction, temperature, and damage mechanisms.
- Misalignment and edge contact. Nonuniform load distribution can sharply increase local pressure.
- Cyclic loading. Cycle count and load spectrum govern contact life.
- Temperature and environment. They affect material properties, lubrication, and surface processes.
Connection with Hertz analysis
Hertz analysis provides the elastic contact-region dimensions $a$ or $b$ and the maximum pressure $p_0$ that form inputs to subsequent strength assessment.
- Identify the initial contact type and local radii of curvature $R_1$ and $R_2$ of the surfaces.
- Check the assumptions of elastic Hertz theory: small contact area, small deformation, smooth surfaces, and no significant plasticity.
- Calculate the reduced elastic modulus $E^*$.
- Determine the reduced curvature or reduced radius $R^*$ for the relevant geometry.
- Use the normal force $F$, or load per unit length $F'=F/L$ for line contact, to determine the dimensions of the contact area or strip.
- Calculate the maximum contact pressure $p_0$ and, where needed, the pressure distribution $p(r)$ or $p(x)$.
- Assess the subsurface stress state and the contact-strength criterion relevant to the component.
- Check cyclic loading, friction, lubrication, roughness, misalignment, and edge effects when they are significant.
Rolling bearings
In rolling bearings, balls or rollers repeatedly load the raceways. Practical bearing-life calculations use specialized standardized load-rating and life relations rather than only comparing $p_0$ with one allowable stress.
Gears
Contact between tooth flanks can locally be approximated as contact of curved bodies. Real gear calculations additionally account for mesh geometry, face-load distribution, dynamics, manufacturing accuracy, lubrication, and factors defined by the selected standard.
Rollers and wheels
For rollers, Hertz line contact and edge effects are important. For wheel-rail contact, normal contact is combined with traction forces, sliding, and a complex cyclic loading history.
Engineering verification procedure
- Determine contact force $F$ and geometry.
- Check applicability of the elastic Hertz model.
- Calculate the contact region and $p_0$.
- Account for actual load distribution, misalignment, and edge effects.
- Identify the expected damage mechanism: plasticity, pitting, wear, scuffing, or another mode.
- Apply a criterion or standard intended for the specific component and operating regime.
- For cyclic contact, verify the required service life separately.
Important limitation
A high Hertz pressure $p_0$ alone is not a complete failure criterion. Contact strength combines the local stress state, properties of the surface layer, cyclic loading, and tribological conditions.